Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 9

Speedup: 2.4×

• 99.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \frac{\left(\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - -1\right) + \frac{0.5}{x \cdot x}}{x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 1.0 (sqrt PI)) (pow (exp (+ x x)) (/ x 2.0)))
  (/
   (+
    (- (/ (+ (/ 1.875 (* x x)) 0.75) (* (* (* x x) x) x)) -1.0)
    (/ 0.5 (* x x)))
   x)))

C

double code(double x) {
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp((x + x)), (x / 2.0))) * ((((((1.875 / (x * x)) + 0.75) / (((x * x) * x) * x)) - -1.0) + (0.5 / (x * x))) / x);
}

Java

public static double code(double x) {
	return ((1.0 / Math.sqrt(Math.PI)) * Math.pow(Math.exp((x + x)), (x / 2.0))) * ((((((1.875 / (x * x)) + 0.75) / (((x * x) * x) * x)) - -1.0) + (0.5 / (x * x))) / x);
}

Python

def code(x):
	return ((1.0 / math.sqrt(math.pi)) * math.pow(math.exp((x + x)), (x / 2.0))) * ((((((1.875 / (x * x)) + 0.75) / (((x * x) * x) * x)) - -1.0) + (0.5 / (x * x))) / x)

Julia

function code(x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(Float64(x + x)) ^ Float64(x / 2.0))) * Float64(Float64(Float64(Float64(Float64(Float64(1.875 / Float64(x * x)) + 0.75) / Float64(Float64(Float64(x * x) * x) * x)) - -1.0) + Float64(0.5 / Float64(x * x))) / x))
end

MATLAB

function tmp = code(x)
	tmp = ((1.0 / sqrt(pi)) * (exp((x + x)) ^ (x / 2.0))) * ((((((1.875 / (x * x)) + 0.75) / (((x * x) * x) * x)) - -1.0) + (0.5 / (x * x))) / x);
end

Wolfram

code[x_] := N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.75), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{e^{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\color{blue}{\left|x\right| \cdot \left|x\right|}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. lift-fabs.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\color{blue}{\left|x\right|} \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. lift-fabs.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \color{blue}{\left|x\right|}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   5. sqr-abs N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\color{blue}{x \cdot x}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   6. exp-prod N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{x}\right)}^{x}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   7. lower-pow.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{x}\right)}^{x}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   8. lower-exp.f64 100.0

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\color{blue}{\left(e^{x}\right)}}^{x}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

4. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{x}\right)}^{x}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
5. Step-by-step derivation

   1. lift-exp.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\color{blue}{\left(e^{x}\right)}}^{x}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{x}\right)}^{x}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   3. sqr-pow N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   4. pow2 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left({\left(e^{x}\right)}^{\left(\frac{x}{2}\right)}\right)}^{2}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   5. pow-exp N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\color{blue}{\left(e^{x \cdot \frac{x}{2}}\right)}}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   6. unpow1 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\color{blue}{{x}^{1}} \cdot \frac{x}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{{x}^{\color{blue}{\left(\frac{2}{2}\right)}} \cdot \frac{x}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   8. sqrt-pow1 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\color{blue}{\sqrt{{x}^{2}}} \cdot \frac{x}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   9. pow2 N/A

      \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\sqrt{\color{blue}{x \cdot x}} \cdot \frac{x}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   10. rem-sqrt-square-rev N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\color{blue}{\left|x\right|} \cdot \frac{x}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   11. unpow1 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\left|x\right| \cdot \frac{\color{blue}{{x}^{1}}}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\left|x\right| \cdot \frac{{x}^{\color{blue}{\left(\frac{2}{2}\right)}}}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   13. sqrt-pow1 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\left|x\right| \cdot \frac{\color{blue}{\sqrt{{x}^{2}}}}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   14. pow2 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\left|x\right| \cdot \frac{\sqrt{\color{blue}{x \cdot x}}}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   15. rem-sqrt-square-rev N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{\left|x\right| \cdot \frac{\color{blue}{\left|x\right|}}{2}}\right)}^{2}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   16. pow-exp N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{e^{\left(\left|x\right| \cdot \frac{\left|x\right|}{2}\right) \cdot 2}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   17. exp-lft-sqr-rev N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(e^{\left|x\right| \cdot \frac{\left|x\right|}{2}} \cdot e^{\left|x\right| \cdot \frac{\left|x\right|}{2}}\right)}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   18. pow-exp N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \left(\color{blue}{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}} \cdot e^{\left|x\right| \cdot \frac{\left|x\right|}{2}}\right)\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   19. pow-exp N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \left({\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)} \cdot \color{blue}{{\left(e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}\right)\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   20. pow-prod-down N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{\left|x\right|} \cdot e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

   21. lower-pow.f64 N/A

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{\left|x\right|} \cdot e^{\left|x\right|}\right)}^{\left(\frac{\left|x\right|}{2}\right)}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

6. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

7. Taylor expanded in x around 0

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \color{blue}{\left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)} \]

9. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \color{blue}{\left(\frac{0.75}{\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x} + \left(\mathsf{fma}\left({x}^{-7}, 1.875, \frac{1}{x}\right) + \frac{0.5}{\left(x \cdot x\right) \cdot x}\right)\right)} \]

10. Taylor expanded in x around -inf

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \left(-1 \cdot \color{blue}{\frac{-1 \cdot \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}{x}}\right) \]

12. Applied rewrites 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \left(-\frac{\left(\left(-\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}\right) - 1\right) - \frac{0.5}{x \cdot x}}{x}\right) \]

13. Final simplification 100.0%

    \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \frac{\left(\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - -1\right) + \frac{0.5}{x \cdot x}}{x} \]
14. Add Preprocessing

Alternative 2: 100.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\sqrt{\pi}}\\ t_1 := \frac{1}{\left|x\right|}\\ \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(t\_1, \frac{\mathsf{fma}\left(1.875, \frac{1}{x \cdot x}, 0.75\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot t\_1\right)}{t\_0 \cdot t\_0} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (sqrt PI))) (t_1 (/ 1.0 (fabs x))))
   (/
    (*
     (exp (* x x))
     (fma
      t_1
      (/ (fma 1.875 (/ 1.0 (* x x)) 0.75) (* (* (* x x) x) x))
      (* (+ (/ 0.5 (* x x)) 1.0) t_1)))
    (* t_0 t_0))))

C

double code(double x) {
	double t_0 = sqrt(sqrt(((double) M_PI)));
	double t_1 = 1.0 / fabs(x);
	return (exp((x * x)) * fma(t_1, (fma(1.875, (1.0 / (x * x)), 0.75) / (((x * x) * x) * x)), (((0.5 / (x * x)) + 1.0) * t_1))) / (t_0 * t_0);
}

Julia

function code(x)
	t_0 = sqrt(sqrt(pi))
	t_1 = Float64(1.0 / abs(x))
	return Float64(Float64(exp(Float64(x * x)) * fma(t_1, Float64(fma(1.875, Float64(1.0 / Float64(x * x)), 0.75) / Float64(Float64(Float64(x * x) * x) * x)), Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) * t_1))) / Float64(t_0 * t_0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Sqrt[Pi], $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(1.875 * N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.75), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, {\left(\left|x\right|\right)}^{-6} \cdot 1.875\right), \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{{x}^{4}}{{\left(\left|x\right|\right)}^{6}}}{{x}^{4}}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

7. Applied rewrites 100.0%

   \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{\mathsf{fma}\left(1.875, \frac{1}{x \cdot x}, 0.75\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]
8. Step-by-step derivation

   1. lift-PI.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\color{blue}{\mathsf{PI}\left(\right)}}} \]

   2. lift-sqrt.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}}} \]

   3. add-sqr-sqrt N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\color{blue}{\sqrt{\mathsf{PI}\left(\right)} \cdot \sqrt{\mathsf{PI}\left(\right)}}}} \]

   4. lift-sqrt.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\color{blue}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   5. lift-PI.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\sqrt{\color{blue}{\pi}} \cdot \sqrt{\mathsf{PI}\left(\right)}}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\sqrt{\pi} \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}}} \]

   7. lift-PI.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\sqrt{\pi} \cdot \sqrt{\color{blue}{\pi}}}} \]

   8. sqrt-prod N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \]

   10. lower-sqrt.f64 N/A

       \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(\frac{15}{8}, \frac{1}{x \cdot x}, \frac{3}{4}\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\color{blue}{\sqrt{\sqrt{\pi}}} \cdot \sqrt{\sqrt{\pi}}} \]

   11. lower-sqrt.f64 100.0

       \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(1.875, \frac{1}{x \cdot x}, 0.75\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\sqrt{\pi}} \cdot \color{blue}{\sqrt{\sqrt{\pi}}}} \]

9. Applied rewrites 100.0%

   \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\mathsf{fma}\left(1.875, \frac{1}{x \cdot x}, 0.75\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\color{blue}{\sqrt{\sqrt{\pi}} \cdot \sqrt{\sqrt{\pi}}}} \]
10. Add Preprocessing

Alternative 3: 100.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \frac{1}{x}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{x}\right) \cdot e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (fma
    (/ (+ (/ 1.875 (* x x)) 0.75) (* (* (* x x) x) x))
    (/ 1.0 x)
    (* (+ (/ 0.5 (* x x)) 1.0) (/ 1.0 x)))
   (exp (* x x)))
  (sqrt PI)))

C

double code(double x) {
	return (fma((((1.875 / (x * x)) + 0.75) / (((x * x) * x) * x)), (1.0 / x), (((0.5 / (x * x)) + 1.0) * (1.0 / x))) * exp((x * x))) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(Float64(Float64(1.875 / Float64(x * x)) + 0.75) / Float64(Float64(Float64(x * x) * x) * x)), Float64(1.0 / x), Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) * Float64(1.0 / x))) * exp(Float64(x * x))) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.75), $MachinePrecision] / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, {\left(\left|x\right|\right)}^{-6} \cdot 1.875\right), \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{{x}^{4}}{{\left(\left|x\right|\right)}^{6}}}{{x}^{4}}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

7. Applied rewrites 100.0%

   \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{\mathsf{fma}\left(1.875, \frac{1}{x \cdot x}, 0.75\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

9. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \frac{1}{x}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{x}\right) \cdot e^{x \cdot x}}{\sqrt{\pi}}} \]
10. Add Preprocessing

Alternative 4: 99.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ e^{x \cdot x} \cdot \frac{\mathsf{fma}\left(\frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \frac{1}{x}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{x}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (exp (* x x))
  (/
   (fma
    (/ 0.75 (* (* (* x x) x) x))
    (/ 1.0 x)
    (* (+ (/ 0.5 (* x x)) 1.0) (/ 1.0 x)))
   (sqrt PI))))

C

double code(double x) {
	return exp((x * x)) * (fma((0.75 / (((x * x) * x) * x)), (1.0 / x), (((0.5 / (x * x)) + 1.0) * (1.0 / x))) / sqrt(((double) M_PI)));
}

Julia

function code(x)
	return Float64(exp(Float64(x * x)) * Float64(fma(Float64(0.75 / Float64(Float64(Float64(x * x) * x) * x)), Float64(1.0 / x), Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) * Float64(1.0 / x))) / sqrt(pi)))
end

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(0.75 / N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * N[(1.0 / x), $MachinePrecision] + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, {\left(\left|x\right|\right)}^{-6} \cdot 1.875\right), \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{\frac{3}{4}}{{x}^{4}}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4}}{{\color{blue}{x}}^{4}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4}}{\color{blue}{{x}^{4}}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   3. metadata-eval N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4}}{{\color{blue}{x}}^{4}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4}}{{x}^{\left(3 + \color{blue}{1}\right)}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   5. pow-plus N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4}}{{x}^{3} \cdot \color{blue}{x}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   6. pow3 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   7. lift-*.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{\frac{3}{4}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

   9. lift-*.f64 99.5

      \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{x}}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

7. Applied rewrites 99.5%

   \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

9. Applied rewrites 99.5%

   \[\leadsto \color{blue}{e^{x \cdot x} \cdot \frac{\mathsf{fma}\left(\frac{0.75}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, \frac{1}{x}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{x}\right)}{\sqrt{\pi}}} \]
10. Add Preprocessing

Alternative 5: 99.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \frac{e^{x \cdot x} \cdot \mathsf{fma}\left({x}^{-7}, 1.875, \frac{\frac{0.5}{x \cdot x} + 1}{x}\right)}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* (exp (* x x)) (fma (pow x -7.0) 1.875 (/ (+ (/ 0.5 (* x x)) 1.0) x)))
  (sqrt PI)))

C

double code(double x) {
	return (exp((x * x)) * fma(pow(x, -7.0), 1.875, (((0.5 / (x * x)) + 1.0) / x))) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) * fma((x ^ -7.0), 1.875, Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) / x))) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[Power[x, -7.0], $MachinePrecision] * 1.875 + N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, {\left(\left|x\right|\right)}^{-6} \cdot 1.875\right), \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}}} \]

5. Taylor expanded in x around 0

   \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{{x}^{4}}{{\left(\left|x\right|\right)}^{6}}}{{x}^{4}}}, \left(\frac{\frac{1}{2}}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

7. Applied rewrites 100.0%

   \[\leadsto \frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \color{blue}{\frac{\mathsf{fma}\left(1.875, \frac{1}{x \cdot x}, 0.75\right)}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}}, \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{e^{x \cdot x} \cdot \color{blue}{\left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)\right)}}{\sqrt{\pi}} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{e^{x \cdot x} \cdot \color{blue}{\mathsf{fma}\left({x}^{-7}, 1.875, \frac{\frac{0.5}{x \cdot x} + 1}{x}\right)}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 6: 99.7% accurate, 3.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left({x}^{-7}, 1.875, \frac{1}{x}\right) \cdot e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (* (fma (pow x -7.0) 1.875 (/ 1.0 x)) (exp (* x x))) (sqrt PI)))

C

double code(double x) {
	return (fma(pow(x, -7.0), 1.875, (1.0 / x)) * exp((x * x))) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(Float64(fma((x ^ -7.0), 1.875, Float64(1.0 / x)) * exp(Float64(x * x))) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[(N[(N[Power[x, -7.0], $MachinePrecision] * 1.875 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, {\left(\left|x\right|\right)}^{-6} \cdot 1.875\right), \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{e^{{x}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{e^{{x}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{\color{blue}{1}}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \color{blue}{e^{{x}^{2}}}}{\sqrt{\pi}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \color{blue}{e^{{x}^{2}}}}{\sqrt{\pi}} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{-7}, 1.875, \frac{1}{x}\right) \cdot e^{x \cdot x}}}{\sqrt{\pi}} \]
8. Add Preprocessing

Alternative 7: 99.7% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{e^{x \cdot x}}{x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ (exp (* x x)) x) (sqrt PI)))

C

double code(double x) {
	return (exp((x * x)) / x) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (Math.exp((x * x)) / x) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (math.exp((x * x)) / x) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / x) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (exp((x * x)) / x) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{e^{x \cdot x} \cdot \mathsf{fma}\left(\frac{1}{\left|x\right|}, \mathsf{fma}\left(0.75, \frac{1}{\left(\left(x \cdot x\right) \cdot x\right) \cdot x}, {\left(\left|x\right|\right)}^{-6} \cdot 1.875\right), \left(\frac{0.5}{x \cdot x} + 1\right) \cdot \frac{1}{\left|x\right|}\right)}{\sqrt{\pi}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{\color{blue}{e^{{x}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{e^{{x}^{2}} \cdot \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{\color{blue}{1}}{{\left(\left|x\right|\right)}^{7}}\right)}{\sqrt{\pi}} \]

   2. *-commutative N/A

      \[\leadsto \frac{\left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \color{blue}{e^{{x}^{2}}}}{\sqrt{\pi}} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{\left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \color{blue}{e^{{x}^{2}}}}{\sqrt{\pi}} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{-7}, 1.875, \frac{1}{x}\right) \cdot e^{x \cdot x}}}{\sqrt{\pi}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{\frac{e^{{x}^{2}}}{\color{blue}{x}}}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{x}}{\sqrt{\pi}} \]

   2. lower-exp.f64 N/A

      \[\leadsto \frac{\frac{e^{{x}^{2}}}{x}}{\sqrt{\pi}} \]

   3. pow2 N/A

      \[\leadsto \frac{\frac{e^{x \cdot x}}{x}}{\sqrt{\pi}} \]

   4. lift-*.f64 99.4

      \[\leadsto \frac{\frac{e^{x \cdot x}}{x}}{\sqrt{\pi}} \]

10. Applied rewrites 99.4%

    \[\leadsto \frac{\frac{e^{x \cdot x}}{\color{blue}{x}}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 8: 1.8% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x \cdot x}}{x \cdot \sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 (* x x)) (* x (sqrt PI))))

C

double code(double x) {
	return (0.5 / (x * x)) / (x * sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (0.5 / (x * x)) / (x * Math.sqrt(Math.PI));
}

Python

def code(x):
	return (0.5 / (x * x)) / (x * math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(0.5 / Float64(x * x)) / Float64(x * sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 / (x * x)) / (x * sqrt(pi));
end

Wolfram

code[x_] := N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]
7. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot 1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2} \cdot 1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   7. rem-sqrt-square-rev N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \sqrt{x \cdot x}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \sqrt{{x}^{2}}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   9. sqrt-pow1 N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot {x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot {x}^{1}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   11. unpow1 N/A

       \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot x} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   13. sqrt-div N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} \cdot \frac{\sqrt{1}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \]

   15. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \]

8. Applied rewrites 1.9%

   \[\leadsto \frac{0.5}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}} \]

   4. pow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot x\right) \cdot \sqrt{\pi}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\color{blue}{\pi}}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\color{blue}{\pi}}\right)} \]

   9. lower-*.f64 1.9

      \[\leadsto \frac{0.5}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\pi}\right)} \]

10. Applied rewrites 1.9%

    \[\leadsto \frac{0.5}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)} \]

12. Applied rewrites 1.9%

    \[\leadsto \frac{\frac{0.5}{x \cdot x}}{x \cdot \color{blue}{\sqrt{\pi}}} \]
13. Add Preprocessing

Alternative 9: 1.8% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x \cdot \left(x \cdot \left(x \cdot \sqrt{\pi}\right)\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (* x (* x (* x (sqrt PI))))))

C

double code(double x) {
	return 0.5 / (x * (x * (x * sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return 0.5 / (x * (x * (x * Math.sqrt(Math.PI))));
}

Python

def code(x):
	return 0.5 / (x * (x * (x * math.sqrt(math.pi))))

Julia

function code(x)
	return Float64(0.5 / Float64(x * Float64(x * Float64(x * sqrt(pi)))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / (x * (x * (x * sqrt(pi))));
end

Wolfram

code[x_] := N[(0.5 / N[(x * N[(x * N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

6. Taylor expanded in x around 0

   \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)} \]
7. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{1}{2} \cdot \left(\frac{1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\color{blue}{\frac{1}{\mathsf{PI}\left(\right)}}}\right) \]

   2. associate-*r* N/A

      \[\leadsto \left(\frac{1}{2} \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{\frac{1}{2} \cdot 1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2} \cdot 1}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \left|x\right|} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   7. rem-sqrt-square-rev N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \sqrt{x \cdot x}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   8. pow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \sqrt{{x}^{2}}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   9. sqrt-pow1 N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot {x}^{\left(\frac{2}{2}\right)}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot {x}^{1}} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   11. unpow1 N/A

       \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot x} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   12. pow2 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \]

   13. sqrt-div N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} \cdot \frac{\sqrt{1}}{\sqrt{\mathsf{PI}\left(\right)}} \]

   14. metadata-eval N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \]

   15. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot x} \cdot \frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \]

8. Applied rewrites 1.9%

   \[\leadsto \frac{0.5}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}}} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}} \]

   4. pow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot x\right) \cdot \sqrt{\pi}} \]

   5. associate-*l* N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)} \]

   6. lower-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)} \]

   7. pow2 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\color{blue}{\pi}}\right)} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\color{blue}{\pi}}\right)} \]

   9. lower-*.f64 1.9

      \[\leadsto \frac{0.5}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\pi}\right)} \]

10. Applied rewrites 1.9%

    \[\leadsto \frac{0.5}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)} \]
11. Step-by-step derivation

    1. lift-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)} \]

    2. lift-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\color{blue}{\pi}}\right)} \]

    3. lift-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\pi}\right)} \]

    4. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

    5. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

    6. associate-*l* N/A

       \[\leadsto \frac{\frac{1}{2}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)}\right)} \]

    8. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{x \cdot \left(x \cdot \left(x \cdot \color{blue}{\sqrt{\mathsf{PI}\left(\right)}}\right)\right)} \]

    9. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{x \cdot \left(x \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)\right)} \]

    10. lift-PI.f64 N/A

        \[\leadsto \frac{\frac{1}{2}}{x \cdot \left(x \cdot \left(x \cdot \sqrt{\pi}\right)\right)} \]

    11. lift-*.f64 1.9

        \[\leadsto \frac{0.5}{x \cdot \left(x \cdot \left(x \cdot \sqrt{\pi}\right)\right)} \]

12. Applied rewrites 1.9%

    \[\leadsto \frac{0.5}{x \cdot \left(x \cdot \color{blue}{\left(x \cdot \sqrt{\pi}\right)}\right)} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025064 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))